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    Separating $L^1$ and $L^2$ on $\RR$ and $(0,1)$

    P-JHUMAY06ANG

    Area
    Real Analysis
    Topics
    Lp Spaces
    Status
    draft

    1. Give a function in \(L^2(\mathbb R)\) that is not in \(L^1(\mathbb R)\).

    2. Give a function in \(L^1((0,1))\) that is not in \(L^2((0,1))\).

    Solution

    For (a), take \[ f(x)=(1+|x|)^{-3/4}. \] Then \[ |f(x)|^2=(1+|x|)^{-3/2}, \] which is integrable on \(\mathbb R\), while \((1+|x|)^{-3/4}\) is not integrable at infinity. Hence \[ f\in L^2(\mathbb R)\setminus L^1(\mathbb R). \]

    For (b), take \[ g(x)=x^{-3/4}, \qquad 0<x<1. \] Since \(3/4<1\), \[ \int_0^1x^{-3/4}\,dx<\infty, \] so \(g\in L^1((0,1))\). But \[ |g(x)|^2=x^{-3/2}, \] and \[ \int_0^1x^{-3/2}\,dx=\infty. \] Thus \[ g\in L^1((0,1))\setminus L^2((0,1)). \]

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